One nice feature of étale cohomology is that it can be used to compute the Betti numbers via a reduction modulo $p$, as for smooth proper schemes $X\to \text{Spec}(\mathbb{Z}[\frac{1}{N}])$ we have isomorphisms $$H^*_{ét}(X_{\bar{\mathbb{F}}_p},\mathbb{Q}_\ell)\cong H^*_{ét}(X_{\mathbb{C}},\mathbb{Q}_\ell)\cong H^*(X(\mathbb{C}),\mathbb{Q}_\ell).$$ One way in which this can be useful if we are interested in characteristic zero phenomena is that the computation of the Betti numbers in postivie characteristic for Flag varieties easier.
My question is if we have similar comparision statements for the trace of an endomorphism? That is to say if we have an endomorphism $f$ of $X$, what can we say about the trace of $f$ on $H^*_{ét}(X_{\bar{\mathbb{F}}_p},\mathbb{Q}_\ell)$ in relation to the trace of $f$ on $H^*_{ét}(X_{\mathbb{C}},\mathbb{Q}_\ell)$?